Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Spring 2024

Speaker:  Yichen Ma, Cornell University
Title: Enumeration of flags in partial orders and convex geometries
Time: 2:30 PM, Monday, March 11, 2024
Place:  Malott 206

Abstract: We consider a Hopf monoid of partial orders and another of convex geometries, and investigate combinatorial invariants constructed from characters on them. Each invariant comes in a pair consisting of a polynomial and a (more general) quasisymmetric function. For partial orders we obtain the order polynomial of Stanley and the enriched order polynomial of Stembridge. For convex geometries we obtain polynomials first introduced by Edelman-Jamison and Billera-Hsiao-Provan. We obtain reciprocity results satisfied by these polynomials from the perspective of characters in a unified manner. We describe the coefficients of the quasisymmetric invariants as enumerating faces on certain simplicial complexes. These include the barycentric subdivision of the CW-sphere of a convex geometry.

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